Large System Analysis of Projection Based Algorithms for the MIMO Broadcast Channel

Large System Analysis of Projection Based Algorithms for the MIMO Broadcast Channel Christian Guthy and Wolfgang Utschick Associate Institute for Signal Processing, Technische Universität München Joint work with Michael L. Honig, Northwestern University, Evanston, IL, USA DFG SPP TakeOFDM, June 22, 2010 Technische Universität München Associate Institute for Signal Processing

Outline 2/19 Introduction System Model Projection Based Algorithms Large System Analysis Simulation Results Conclusions Publications

Introduction 3/19 Analytical performance evaluation of algorithms requiring full transmit CSI in the MIMO OFDM Broadcast Channel (BC) Analytical expressions for average sum rate difficult to obtain for those algorithms Large system analysis Number of transmit antennas and receive antennas goes towards at finite fixed ratio β deterministic behavior of eigenvalues of large random matrices analytical expressions for average sum rates can be obtained Use large system results as approximations in finite systems with same β

System Model 4/19 MIMO OFDM broadcast channel with C carriers and K users with M Rx receive antennas M Tx transmit antennas Full CSI at transmitter H k,c C M Rx M Tx : channel matrix of user k on carrier c with Gaussian i.i.d. entries vec[h k,c ] CN (0,I MTx M Rx ) Sum transmit power constraint P Tx Additive white Gaussian noise with unit variance

Projection Based Algorithms 5/19 Block Diagonalization (BD) [Spencer et al. 2004] Block Diagonalization with Dirty Paper Coding (BD-DPC) [Stankovic, Haardt 2004], [Tejera et al. 2005] Successive Encoding Successive Allocation Method (SESAM) [Tejera et al. 2006]

Projection Based Algorithms Decomposition of the MIMO broadcast channel on each carrier into a system of N = min(km Rx,M Tx ) scalar interference free subchannels: 6/19 Channel gains: λ i,c = gi,ch H πc(i),ct i,c 2 2 = ( ) 1 M Rx ρ n(i) H πc(i),cp i,c Hπ H 1 c(i),c = M Rx λ i,c MRx MRx P i,c = V i,c V H i,c : projection matrix, V i,c orthonormal g i,c, t i,c : receive / transmit filter for i-th data stream on carrier c ρ n(i) (A): n(i)-th strongest eigenvalue of A User allocation: π c : {1,...,N} {1,...,K},i π c (i) Encoding order: data stream on subchannel i encoded at i-th place ( DPC) Subchannel powers γ i,c

Projection Based Algorithms Decomposition of the MIMO broadcast channel on each carrier into a system of N = min(km Rx,M Tx ) scalar interference free subchannels: 6/19 s 1,c γ1,c MRx λ 1,c ŝ 1,c. η 1,c s i,c ŝ i,c γi,c MRx λ i,c. η i,c s N,c ŝ N,c γn,c MRx λ N,c η N,c

SESAM Successive subchannel-wise user allocation: 7/19 For sum rate maximization: π c (i) = argmaxρ 1 (H k,c P i,c,sesam Hk,c) H k Transmit and receive filter computation: {t i,c,g i,c } = argmaxg H H k,c P i,c,sesam t, s.t. g H g = 1,t H t = 1 t,g P i,c,sesam project into null{g H 1,c H π c(1),c,...,g H i 1,c H π c(i 1),c} Interference suppression through DPC and linear transmit and receive signal processing n(i) = 1

Outline 8/19 Introduction System Model Projection Based Algorithms Large System Analysis Simulation Results Conclusions Publications

Large System Analysis 9/19 M Tx, M Rx, M Tx M Rx From random matrix theory: = β, finite, K finite, P Tx finite Empirical distribution of the eigenvalues ρ i of a N N matrix A F N A (x) = 1 N {ρ i ρ i x;i = 1,...,N} converges to asymptotic limit for N for many kind of random matrices Asymptotic eigenvalue distribution (a.e.d.) (x) = d dx lim N FN A (x)

Large System Analysis 9/19 M Tx, M Rx, M Tx M Rx = β, finite, K finite, P Tx finite Here: Does empirical distribution of channel gains F N alg (x) = 1 N {λ i,c λ i,c x;i = 1,...,N,c = 1,...,C} converge to asymptotic limit? Does asymptotic distribution of channel gains exist? alg (x) = d dx lim N FN alg(x)

Large System Analysis of BD Approaches 10/19 BD-DPC Kactive BD-DPC (λ) = j=1 [λ a(βbd-dpc (j))] + [b(β BD-DPC (j)) λ] + 2πλK active β BD-DPC (j) = β (j 1), a(x) = (1 x) 2, b(x) = (1 + x) 2 [x] + = max(0,x), K active = min ( β,k) BD BD (λ) = [λ a(βbd )] + [b(β BD ) λ] + 2πλ Same β BD = β (K active 1) for all users

Large System Analysis of SESAM 11/19 Finite System: N finite = min(,km Rx,finite ) allocation steps per carrier λ i,c = ρ 1 (H πc(i),cp i,c,sesam H H π c(i),c ) Large System: N finite allocation steps per carrier M Tx subchannels allocated at once per step with channel gains ρ 1 (H πc(i),cp ( ) i,c,sesam HH π c(i),c),..., ρ M Tx (H πc(i),cp ( ) i,,csesam HH π c(i),c) Basis of projector: V i,c,sesam C (i 1) Basis of projector: V ( ) i,c,sesam M CM Tx M Tx (i 1) Tx Ratio #columns #rows = 1 (i 1) identical for V ( ) i,c,sesam and V i,c,sesam

Large System Analysis of SESAM SESAM (λ) = C c=1 M Tx,finite i=1 { 1 C i,c (λ)i 1, λ > ˆλ i,c i,c(λ), I i,c (λ) = 0, else 12/19 ˆλ i,c i,c (λ)dλ = 1 ( ) i,c (λ) are tails of a.e.d. of P i,c,sesam HH π H c(i),c π c(i),cp ( ) i,c,sesam Illustration:

Large System Analysis of SESAM Computation of i,c (λ): A priori fixed user allocation lower bound for sum rate 13/19 1. First steps (no subchannel assigned to same user in previous steps on same carrier): V ( ) i,c,sesam independent of channel matrices H π c(i),c Hπc(i),cV ( ) i,c,sesam CM Rx M Tx (i 1)M Tx / contains Gaussian i.i.d. entries with zero mean and unit variance Marčenko Pastur : i,c (x) = [λ a(βsesam (i))] + [b(β SESAM (i)) λ] + 2πλ β SESAM (i) = β(1 (i 1)/ ), a(x) = (1 x) 2, b(x) = (1+ x) 2 [x] + = max(0,x)

Large System Analysis of SESAM Computation of i,c (λ): A priori fixed user allocation lower bound for sum rate 13/19 2. Further steps: f i,c (λ) = ( 1 (i 1) ) B M i,c (λ) + (i 1) δ(λ) Tx,finite B i,c (λ): a.e.d. of matrix B i,c = V ( ),H i,c,sesam HH π c(i),c H π c(i),cv ( ) i,c,sesam Stieltjes transform S Bi,c (z) of matrix B i,c β i,c = n1 0 B (λ) i lc,i,c 1 β i,c + (λ z) β i,c S Bi,c (z) dλ = (i l c,i ) 1 (i l c,i ) M n Tx,finite (i 1) (i l, 1 c,i) 0 B i lc,i,c (λ)dλ = (i l c,i) 1 (i l c,i) l c,i : last step before step i, where user π c (i) received a subchannel

Sum Rate Computation Objective: Maximization of sum rate 14/19 Finite system: R sum = C N log 2 (1+γ i M Rx λ i,c ) = c=1 i=1 C N max ( 0, [log 2 (ηm Rx ) + log 2 (λ i,c )] ) c=1 i=1 Large System (N ): R sum N λ lb Water-filling in the large system limit: [ log2 ( MRx η ( )) + log 2 (λ) ] alg (λ)dλ M Rx η ( ) = P Tx + N 1 M Rx λ lb λ alg (λ)dλ N M Rx λ lb alg (λ)dλ η ( ) = 1 λ lb

Outline 15/19 Introduction System Model Projection Based Algorithms Large System Analysis Simulation Results Conclusions Publications

Simulation Results 16/19 K = 5 users, β = M Tx M Rx = 2, P Tx = 10 9 8 Normalized sum rate (bpcu) 7 6 5 4 3 Average sum capacity Average SESAM succ UA Average SESAM fixed UA Large system SESAM 0 10 20 30 40 Number of transmit antennas

Simulation Results 16/19 K = 5 users, β = M Tx M Rx = 2, P Tx = 10 9 8 Normalized sum rate (bpcu) 7 6 5 4 3 2 Average sum capacity Large system SESAM Average BD DPC succ UA 1 Average BD DPC fixed UA Large system BD DPC 0 0 10 20 30 40 Number of transmit antennas

Simulation Results 16/19 K = 5 users, β = M Tx M Rx = 2, P Tx = 10 9 8 Normalized sum rate (bpcu) 7 6 5 4 3 2 Average sum capacity Large system SESAM Large system BD DPC 1 Average BD Large system BD 0 0 10 20 30 40 Number of transmit antennas

Conclusions 17/19 Analytical expression for lower bound of sum rate of SESAM, BD-DPC and BD based on large system analysis Good estimation for average sum rate also in finite systems Results applicable to many Quality of Service (QoS) constrained problems

Publications 18/19 [1] C. Guthy, W. Utschick, and M.L Honig: "Large System Analysis of Projection Based Algorithms for the MIMO Broadcast Channel", Proc. of IEEE International Symposium on Information Theory (ISIT), June 2010 [2] C. Guthy, W. Utschick, and M.L Honig: "Large System Analysis of the Successive Encoding Successive Allocation Method for the MIMO BC", Proc. of ITG Workshop on Smart Antennas (WSA), February 2010 [3] C. Guthy, W. Utschick, R. Hunger and M. Joham: "Efficient Weighted Sum Rate Maximization with Linear Precoding", IEEE Transactions on Signal Processing, April 2010 [4] C. Guthy, W. Utschick, R. Hunger and M. Joham: "Weighted Sum Rate Maximization in the MIMO MAC with Linear Transceivers: Algorithmic Solutions", Asilomar Conference on Signals, Systems, and Computers, November 2009 [5] R. Hunger, M. Joham, C. Guthy, and W. Utschick: "Weighted Sum Rate Maximization in the MIMO MAC with Linear Transceivers: Asymptotic Results", Asilomar Conference on Signals, Systems, and Computers, November 2009

Publications 19/19 [6] C. Guthy, W. Utschick, and G. Dietl: "Low Complexity Linear Zero-Forcing for the MIMO Broadcast Channel ", IEEE Journal on Selected Topics in Signal Processing, December 2009 [7] C. Guthy, W. Utschick, and G. Dietl: "Spatial Resource Allocation for the Multiuser Multicarrier MIMO Brodacast Channel A QoS Optimization Perspective", Proc. of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), March 2010